10 112

10_111

10_113

(KnotPlot image)

See the full Rolfsen Knot Table.

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Visit 10 112 at Knotilus!

[edit 10 112 Quick Notes]

[edit 10 112 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₃₉₄ X_18,11,19,12 X_20,13,1,14 X_2,16,3,15 X_4,17,5,18 X_12,19,13,20 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9
Gauss code	1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4
Dowker-Thistlethwaite code	6 8 10 14 16 18 20 2 4 12
Conway Notation	[8*3]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{3, 12}, {2, 10}, {4, 11}, {5, 3}, {6, 4}, {7, 5}, {1, 6}, {9, 2}, {10, 8}, {12, 9}, {11, 7}, {8, 1}]

[edit Notes on presentations of 10 112]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 112"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₆₂₇₁ X₈₃₉₄ X_18,11,19,12 X_20,13,1,14 X_2,16,3,15 X_4,17,5,18 X_12,19,13,20 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9

In[5]:= GaussCode[K]

Out[5]= 1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4

In[6]:= DTCode[K]

Out[6]= 6 8 10 14 16 18 20 2 4 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [8*3]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{-1,-1,-1,2,-1,2,-1,2,-1,2\})}

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { 3, 10, 3 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{3, 12}, {2, 10}, {4, 11}, {5, 3}, {6, 4}, {7, 5}, {1, 6}, {9, 2}, {10, 8}, {12, 9}, {11, 7}, {8, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-3]
Hyperbolic Volume	14.7559
A-Polynomial	See Data:10 112/A-polynomial

[edit Notes for 10 112's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4}
Rasmussen s-Invariant	2

[edit Notes for 10 112's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }
Conway polynomial	$-z^{8}-3z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 87, -2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-4 q^2+7 q-10+14 q^{-1} -14 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-7 a^2 z^4+3 z^4+a^4 z^2+z^2-2 a^4+4 a^2-1}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 a^3 z^9+3 a z^9+7 a^4 z^8+13 a^2 z^8+6 z^8+8 a^5 z^7+4 a^3 z^7+4 z^7 a^{-1} +7 a^6 z^6-9 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -18 z^6+4 a^7 z^5-8 a^5 z^5-17 a^3 z^5-16 a z^5-11 z^5 a^{-1} +a^8 z^4-7 a^6 z^4+3 a^4 z^4+28 a^2 z^4-2 z^4 a^{-2} +15 z^4-3 a^7 z^3-a^5 z^3+9 a^3 z^3+13 a z^3+6 z^3 a^{-1} +a^6 z^2+a^4 z^2-3 a^2 z^2-3 z^2+2 a^5 z+2 a^3 z-2 a^4-4 a^2-1}
The A2 invariant	$q^{20}-2q^{18}+q^{16}-3q^{14}-q^{12}+2q^{10}-q^{8}+6q^{6}-q^{4}+3q^{2}-2q^{-2}+q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+9 q^{106}-6 q^{104}-2 q^{102}+18 q^{100}-32 q^{98}+47 q^{96}-50 q^{94}+34 q^{92}-6 q^{90}-34 q^{88}+75 q^{86}-104 q^{84}+117 q^{82}-101 q^{80}+50 q^{78}+25 q^{76}-110 q^{74}+182 q^{72}-205 q^{70}+161 q^{68}-63 q^{66}-68 q^{64}+177 q^{62}-218 q^{60}+166 q^{58}-45 q^{56}-99 q^{54}+184 q^{52}-176 q^{50}+58 q^{48}+108 q^{46}-239 q^{44}+275 q^{42}-192 q^{40}+16 q^{38}+182 q^{36}-322 q^{34}+358 q^{32}-271 q^{30}+102 q^{28}+103 q^{26}-251 q^{24}+316 q^{22}-264 q^{20}+137 q^{18}+29 q^{16}-166 q^{14}+221 q^{12}-170 q^{10}+47 q^8+109 q^6-214 q^4+212 q^2-106-64 q^{-2} +214 q^{-4} -286 q^{-6} +244 q^{-8} -112 q^{-10} -54 q^{-12} +184 q^{-14} -238 q^{-16} +206 q^{-18} -112 q^{-20} +3 q^{-22} +71 q^{-24} -104 q^{-26} +94 q^{-28} -58 q^{-30} +25 q^{-32} +5 q^{-34} -17 q^{-36} +17 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 10_112.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-3 q^{13}+3 q^{11}-4 q^9+3 q^7+4 q-3 q^{-1} +3 q^{-3} -3 q^{-5} + q^{-7} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-3 q^{40}+8 q^{36}-10 q^{34}+q^{32}+19 q^{30}-24 q^{28}-5 q^{26}+34 q^{24}-24 q^{22}-19 q^{20}+32 q^{18}-3 q^{16}-22 q^{14}+9 q^{12}+19 q^{10}-12 q^8-17 q^6+28 q^4+4 q^2-33+23 q^{-2} +18 q^{-4} -34 q^{-6} +5 q^{-8} +23 q^{-10} -17 q^{-12} -7 q^{-14} +12 q^{-16} - q^{-18} -3 q^{-20} + q^{-22} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{81}-3 q^{79}+5 q^{75}+2 q^{73}-6 q^{71}-6 q^{69}+12 q^{67}-5 q^{65}-17 q^{63}+17 q^{61}+41 q^{59}-34 q^{57}-85 q^{55}+41 q^{53}+149 q^{51}-16 q^{49}-209 q^{47}-40 q^{45}+238 q^{43}+117 q^{41}-225 q^{39}-182 q^{37}+154 q^{35}+224 q^{33}-74 q^{31}-218 q^{29}-21 q^{27}+189 q^{25}+96 q^{23}-142 q^{21}-152 q^{19}+96 q^{17}+191 q^{15}-48 q^{13}-219 q^{11}+q^9+241 q^7+53 q^5-238 q^3-119 q+221 q^{-1} +179 q^{-3} -157 q^{-5} -228 q^{-7} +78 q^{-9} +231 q^{-11} +10 q^{-13} -196 q^{-15} -81 q^{-17} +129 q^{-19} +109 q^{-21} -58 q^{-23} -95 q^{-25} +5 q^{-27} +62 q^{-29} +19 q^{-31} -30 q^{-33} -15 q^{-35} +7 q^{-37} +8 q^{-39} - q^{-41} -3 q^{-43} + q^{-45} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{132}-3 q^{130}+5 q^{126}-q^{124}+6 q^{122}-13 q^{120}-7 q^{118}+5 q^{116}-8 q^{114}+43 q^{112}+4 q^{110}-5 q^{108}-39 q^{106}-115 q^{104}+56 q^{102}+140 q^{100}+191 q^{98}-44 q^{96}-479 q^{94}-272 q^{92}+239 q^{90}+819 q^{88}+495 q^{86}-795 q^{84}-1234 q^{82}-420 q^{80}+1352 q^{78}+1806 q^{76}-107 q^{74}-1950 q^{72}-1927 q^{70}+623 q^{68}+2626 q^{66}+1469 q^{64}-1162 q^{62}-2728 q^{60}-1001 q^{58}+1753 q^{56}+2269 q^{54}+494 q^{52}-1872 q^{50}-1863 q^{48}+88 q^{46}+1660 q^{44}+1462 q^{42}-416 q^{40}-1605 q^{38}-1000 q^{36}+713 q^{34}+1600 q^{32}+521 q^{30}-1187 q^{28}-1500 q^{26}+174 q^{24}+1659 q^{22}+1136 q^{20}-980 q^{18}-2018 q^{16}-376 q^{14}+1728 q^{12}+1941 q^{10}-397 q^8-2381 q^6-1386 q^4+1078 q^2+2541+905 q^{-2} -1742 q^{-4} -2211 q^{-6} -471 q^{-8} +1943 q^{-10} +1957 q^{-12} -71 q^{-14} -1709 q^{-16} -1670 q^{-18} +288 q^{-20} +1549 q^{-22} +1148 q^{-24} -202 q^{-26} -1332 q^{-28} -801 q^{-30} +243 q^{-32} +887 q^{-34} +646 q^{-36} -264 q^{-38} -566 q^{-40} -377 q^{-42} +116 q^{-44} +400 q^{-46} +164 q^{-48} -55 q^{-50} -195 q^{-52} -101 q^{-54} +59 q^{-56} +62 q^{-58} +41 q^{-60} -20 q^{-62} -29 q^{-64} - q^{-66} +3 q^{-68} +8 q^{-70} - q^{-72} -3 q^{-74} + q^{-76} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{195}-3 q^{193}+5 q^{189}-q^{187}+3 q^{185}-q^{183}-14 q^{181}-14 q^{179}+8 q^{177}+26 q^{175}+39 q^{173}+27 q^{171}-44 q^{169}-119 q^{167}-111 q^{165}+46 q^{163}+233 q^{161}+297 q^{159}+89 q^{157}-396 q^{155}-732 q^{153}-406 q^{151}+522 q^{149}+1377 q^{147}+1244 q^{145}-325 q^{143}-2358 q^{141}-2807 q^{139}-538 q^{137}+3226 q^{135}+5261 q^{133}+2790 q^{131}-3383 q^{129}-8363 q^{127}-6776 q^{125}+1819 q^{123}+11098 q^{121}+12328 q^{119}+2363 q^{117}-12050 q^{115}-18353 q^{113}-9116 q^{111}+9844 q^{109}+22819 q^{107}+17308 q^{105}-3982 q^{103}-23854 q^{101}-24722 q^{99}-4508 q^{97}+20466 q^{95}+29011 q^{93}+13444 q^{91}-13220 q^{89}-28708 q^{87}-20448 q^{85}+4095 q^{83}+24112 q^{81}+23630 q^{79}+4488 q^{77}-16577 q^{75}-22887 q^{73}-10738 q^{71}+8556 q^{69}+19145 q^{67}+13802 q^{65}-1675 q^{63}-14248 q^{61}-14346 q^{59}-2957 q^{57}+9853 q^{55}+13501 q^{53}+5484 q^{51}-6920 q^{49}-12671 q^{47}-6700 q^{45}+5592 q^{43}+12784 q^{41}+7707 q^{39}-5350 q^{37}-14066 q^{35}-9442 q^{33}+5142 q^{31}+16192 q^{29}+12420 q^{27}-3966 q^{25}-18267 q^{23}-16515 q^{21}+1026 q^{19}+19110 q^{17}+21012 q^{15}+3968 q^{13}-17673 q^{11}-24717 q^9-10372 q^7+13293 q^5+26080 q^3+17093 q-6169 q^{-1} -24159 q^{-3} -22158 q^{-5} -2447 q^{-7} +18441 q^{-9} +24012 q^{-11} +10687 q^{-13} -10037 q^{-15} -21692 q^{-17} -16237 q^{-19} +756 q^{-21} +15621 q^{-23} +17721 q^{-25} +6925 q^{-27} -7576 q^{-29} -15018 q^{-31} -11105 q^{-33} -52 q^{-35} +9478 q^{-37} +11239 q^{-39} +5213 q^{-41} -3369 q^{-43} -8312 q^{-45} -6957 q^{-47} -1275 q^{-49} +4158 q^{-51} +5888 q^{-53} +3506 q^{-55} -666 q^{-57} -3504 q^{-59} -3462 q^{-61} -1239 q^{-63} +1192 q^{-65} +2281 q^{-67} +1653 q^{-69} +133 q^{-71} -1007 q^{-73} -1162 q^{-75} -552 q^{-77} +181 q^{-79} +572 q^{-81} +454 q^{-83} +82 q^{-85} -172 q^{-87} -215 q^{-89} -108 q^{-91} +16 q^{-93} +82 q^{-95} +55 q^{-97} + q^{-99} -19 q^{-101} -15 q^{-103} -5 q^{-105} +3 q^{-107} +8 q^{-109} - q^{-111} -3 q^{-113} + q^{-115} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{270}-3 q^{268}+5 q^{264}-q^{262}+3 q^{260}-4 q^{258}-2 q^{256}-21 q^{254}-11 q^{252}+42 q^{250}+28 q^{248}+36 q^{246}-3 q^{244}-55 q^{242}-157 q^{240}-127 q^{238}+107 q^{236}+225 q^{234}+326 q^{232}+190 q^{230}-160 q^{228}-734 q^{226}-821 q^{224}-93 q^{222}+758 q^{220}+1583 q^{218}+1449 q^{216}+33 q^{214}-2441 q^{212}-3745 q^{210}-2161 q^{208}+1426 q^{206}+5785 q^{204}+7359 q^{202}+3530 q^{200}-5400 q^{198}-13415 q^{196}-13431 q^{194}-3589 q^{192}+13408 q^{190}+26609 q^{188}+24074 q^{186}+1478 q^{184}-29679 q^{182}-48096 q^{180}-37772 q^{178}+4186 q^{176}+55610 q^{174}+81157 q^{172}+54349 q^{170}-17776 q^{168}-93662 q^{166}-122736 q^{164}-71743 q^{162}+39507 q^{160}+145090 q^{158}+169711 q^{156}+83551 q^{154}-71025 q^{152}-202254 q^{150}-215345 q^{148}-87509 q^{146}+112381 q^{144}+257605 q^{142}+248375 q^{140}+79265 q^{138}-154324 q^{136}-301417 q^{134}-263819 q^{132}-57139 q^{130}+189978 q^{128}+322701 q^{126}+256764 q^{124}+29559 q^{122}-212564 q^{120}-320019 q^{118}-227969 q^{116}-1781 q^{114}+216152 q^{112}+293743 q^{110}+188732 q^{108}-21342 q^{106}-204151 q^{104}-250290 q^{102}-145830 q^{100}+36003 q^{98}+180150 q^{96}+202898 q^{94}+104568 q^{92}-45655 q^{90}-151559 q^{88}-157267 q^{86}-67675 q^{84}+52872 q^{82}+127501 q^{80}+115629 q^{78}+31904 q^{76}-62677 q^{74}-108267 q^{72}-76318 q^{70}+5389 q^{68}+78347 q^{66}+90646 q^{64}+34071 q^{62}-48065 q^{60}-95505 q^{58}-68472 q^{56}+14805 q^{54}+95245 q^{52}+107948 q^{50}+36161 q^{48}-72932 q^{46}-140329 q^{44}-108342 q^{42}+8598 q^{40}+135384 q^{38}+176663 q^{36}+93826 q^{34}-65353 q^{32}-193987 q^{30}-197546 q^{28}-65014 q^{26}+124891 q^{24}+241972 q^{22}+202838 q^{20}+25026 q^{18}-178074 q^{16}-272696 q^{14}-194117 q^{12}+14895 q^{10}+218361 q^8+286782 q^6+174267 q^4-47186 q^2-239543-284538 q^{-2} -153010 q^{-4} +67717 q^{-6} +243003 q^{-8} +267322 q^{-10} +134083 q^{-12} -72869 q^{-14} -229033 q^{-16} -243101 q^{-18} -118385 q^{-20} +66287 q^{-22} +199949 q^{-24} +213825 q^{-26} +107454 q^{-28} -49879 q^{-30} -164671 q^{-32} -180650 q^{-34} -97700 q^{-36} +27443 q^{-38} +126436 q^{-40} +147194 q^{-42} +88688 q^{-44} -7493 q^{-46} -88419 q^{-48} -113621 q^{-50} -79365 q^{-52} -8326 q^{-54} +55890 q^{-56} +83122 q^{-58} +66022 q^{-60} +19031 q^{-62} -29776 q^{-64} -57423 q^{-66} -51503 q^{-68} -22737 q^{-70} +12115 q^{-72} +35303 q^{-74} +37728 q^{-76} +21839 q^{-78} -2141 q^{-80} -19462 q^{-82} -24867 q^{-84} -17354 q^{-86} -3304 q^{-88} +9548 q^{-90} +15028 q^{-92} +11770 q^{-94} +4397 q^{-96} -3622 q^{-98} -7855 q^{-100} -7459 q^{-102} -3406 q^{-104} +1041 q^{-106} +3498 q^{-108} +3919 q^{-110} +2199 q^{-112} +41 q^{-114} -1561 q^{-116} -1718 q^{-118} -1042 q^{-120} -227 q^{-122} +523 q^{-124} +687 q^{-126} +471 q^{-128} +48 q^{-130} -147 q^{-132} -200 q^{-134} -164 q^{-136} -25 q^{-138} +53 q^{-140} +74 q^{-142} +15 q^{-144} +2 q^{-146} -5 q^{-148} -19 q^{-150} -5 q^{-152} +3 q^{-154} +8 q^{-156} - q^{-158} -3 q^{-160} + q^{-162} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-2 q^{18}+q^{16}-3 q^{14}-q^{12}+2 q^{10}-q^8+6 q^6-q^4+3 q^2-2 q^{-2} + q^{-4} -2 q^{-6} + q^{-8} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+69 q^{52}-120 q^{50}+190 q^{48}-268 q^{46}+357 q^{44}-460 q^{42}+576 q^{40}-686 q^{38}+790 q^{36}-878 q^{34}+906 q^{32}-846 q^{30}+671 q^{28}-372 q^{26}-56 q^{24}+558 q^{22}-1086 q^{20}+1554 q^{18}-1926 q^{16}+2140 q^{14}-2158 q^{12}+2014 q^{10}-1680 q^8+1236 q^6-688 q^4+132 q^2+388-822 q^{-2} +1101 q^{-4} -1220 q^{-6} +1182 q^{-8} -1040 q^{-10} +827 q^{-12} -590 q^{-14} +384 q^{-16} -226 q^{-18} +117 q^{-20} -52 q^{-22} +20 q^{-24} -6 q^{-26} + q^{-28} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{52}-2 q^{50}-q^{48}+3 q^{46}-3 q^{44}+2 q^{42}+6 q^{40}-q^{38}-3 q^{36}+2 q^{34}+8 q^{32}-11 q^{30}-15 q^{28}+7 q^{26}+q^{24}-15 q^{22}+2 q^{20}+17 q^{18}-q^{16}-2 q^{14}+9 q^{12}+6 q^{10}-7 q^8+q^6+14 q^4-9 q^2-6+11 q^{-2} -13 q^{-6} - q^{-8} +9 q^{-10} - q^{-12} -7 q^{-14} +2 q^{-16} +6 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-24} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+8 q^{42}-9 q^{40}-3 q^{38}+18 q^{36}-13 q^{34}-11 q^{32}+28 q^{30}-12 q^{28}-18 q^{26}+24 q^{24}-11 q^{22}-18 q^{20}+7 q^{18}+3 q^{16}-q^{14}-3 q^{12}+19 q^{10}+14 q^8-19 q^6+11 q^4+14 q^2-28+4 q^{-2} +15 q^{-4} -21 q^{-6} +7 q^{-8} +9 q^{-10} -10 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} + q^{-20} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-2 q^{23}+2 q^{21}-5 q^{19}+q^{17}-4 q^{15}+2 q^{13}+q^{11}+4 q^9+4 q^7+q^5+3 q^3-3 q+2 q^{-1} -4 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-6 q^{76}+15 q^{74}-17 q^{72}-3 q^{70}+45 q^{68}-86 q^{66}+84 q^{64}-10 q^{62}-112 q^{60}+210 q^{58}-204 q^{56}+69 q^{54}+154 q^{52}-343 q^{50}+362 q^{48}-147 q^{46}-237 q^{44}+591 q^{42}-689 q^{40}+425 q^{38}+131 q^{36}-695 q^{34}+991 q^{32}-892 q^{30}+417 q^{28}+108 q^{26}-527 q^{24}+596 q^{22}-431 q^{20}+217 q^{18}-87 q^{16}+239 q^{14}-424 q^{12}+582 q^{10}-413 q^8-31 q^6+577 q^4-987 q^2+1018-706 q^{-2} +167 q^{-4} +339 q^{-6} -639 q^{-8} +659 q^{-10} -445 q^{-12} +140 q^{-14} +113 q^{-16} -223 q^{-18} +197 q^{-20} -104 q^{-22} +17 q^{-24} +26 q^{-26} -29 q^{-28} +17 q^{-30} -6 q^{-32} + q^{-34} }

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{58}-2 q^{56}-2 q^{54}+6 q^{52}-2 q^{50}-9 q^{48}+11 q^{46}+10 q^{44}-17 q^{42}+q^{40}+20 q^{38}-5 q^{36}-19 q^{34}+9 q^{32}+7 q^{30}-28 q^{28}-15 q^{26}+17 q^{24}-12 q^{22}-20 q^{20}+34 q^{18}+19 q^{16}-11 q^{14}+20 q^{12}+30 q^{10}-13 q^8-14 q^6+8 q^4-21-4 q^{-2} +13 q^{-4} -5 q^{-6} -3 q^{-8} +9 q^{-10} -5 q^{-14} +4 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{30}-2 q^{28}+2 q^{26}-4 q^{24}-q^{22}-2 q^{20}-3 q^{18}+2 q^{16}+6 q^{12}+2 q^{10}+6 q^8+3 q^4-2 q^2-1-3 q^{-4} +2 q^{-6} -2 q^{-8} + q^{-10} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-3 q^{46}+6 q^{44}-10 q^{42}+15 q^{40}-21 q^{38}+26 q^{36}-31 q^{34}+33 q^{32}-32 q^{30}+24 q^{28}-14 q^{26}-2 q^{24}+17 q^{22}-34 q^{20}+49 q^{18}-59 q^{16}+67 q^{14}-63 q^{12}+59 q^{10}-44 q^8+31 q^6-11 q^4-4 q^2+18-28 q^{-2} +33 q^{-4} -35 q^{-6} +31 q^{-8} -27 q^{-10} +20 q^{-12} -13 q^{-14} +7 q^{-16} -3 q^{-18} + q^{-20} }

1,0

q^{78}-3q^{74}-3q^{72}+3q^{70}+9q^{68}+2q^{66}-12q^{64}-12q^{62}+8q^{60}+22q^{58}+6q^{56}-23q^{54}-22q^{52}+13q^{50}+34q^{48}+3q^{46}-33q^{44}-19q^{42}+23q^{40}+25q^{38}-15q^{36}-31q^{34}+q^{32}+26q^{30}+3q^{28}-24q^{26}-9q^{24}+22q^{22}+15q^{20}-13q^{18}-13q^{16}+19q^{14}+24q^{12}-8q^{10}-29q^{8}+q^{6}+33q^{4}+15q^{2}-30-32q^{-2}+13q^{-4}+35q^{-6}+3q^{-8}-31q^{-10}-17q^{-12}+18q^{-14}+21q^{-16}-5q^{-18}-15q^{-20}-2q^{-22}+9q^{-24}+4q^{-26}-3q^{-28}-3q^{-30}+q^{-34}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+9 q^{58}-12 q^{56}+12 q^{54}-15 q^{52}+22 q^{50}-23 q^{48}+21 q^{46}-25 q^{44}+29 q^{42}-21 q^{40}+15 q^{38}-13 q^{36}+3 q^{34}+6 q^{32}-20 q^{30}+18 q^{28}-39 q^{26}+41 q^{24}-46 q^{22}+51 q^{20}-48 q^{18}+56 q^{16}-35 q^{14}+45 q^{12}-27 q^{10}+23 q^8-9 q^6+q^4+3 q^2-18+20 q^{-2} -26 q^{-4} +25 q^{-6} -28 q^{-8} +28 q^{-10} -23 q^{-12} +20 q^{-14} -16 q^{-16} +12 q^{-18} -7 q^{-20} +4 q^{-22} -3 q^{-24} + q^{-26} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+9 q^{106}-6 q^{104}-2 q^{102}+18 q^{100}-32 q^{98}+47 q^{96}-50 q^{94}+34 q^{92}-6 q^{90}-34 q^{88}+75 q^{86}-104 q^{84}+117 q^{82}-101 q^{80}+50 q^{78}+25 q^{76}-110 q^{74}+182 q^{72}-205 q^{70}+161 q^{68}-63 q^{66}-68 q^{64}+177 q^{62}-218 q^{60}+166 q^{58}-45 q^{56}-99 q^{54}+184 q^{52}-176 q^{50}+58 q^{48}+108 q^{46}-239 q^{44}+275 q^{42}-192 q^{40}+16 q^{38}+182 q^{36}-322 q^{34}+358 q^{32}-271 q^{30}+102 q^{28}+103 q^{26}-251 q^{24}+316 q^{22}-264 q^{20}+137 q^{18}+29 q^{16}-166 q^{14}+221 q^{12}-170 q^{10}+47 q^8+109 q^6-214 q^4+212 q^2-106-64 q^{-2} +214 q^{-4} -286 q^{-6} +244 q^{-8} -112 q^{-10} -54 q^{-12} +184 q^{-14} -238 q^{-16} +206 q^{-18} -112 q^{-20} +3 q^{-22} +71 q^{-24} -104 q^{-26} +94 q^{-28} -58 q^{-30} +25 q^{-32} +5 q^{-34} -17 q^{-36} +17 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 112"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }

In[5]:= Conway[K][z]

Out[5]= $-z^{8}-3z^{6}-z^{4}+2z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 87, -2 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-4 q^2+7 q-10+14 q^{-1} -14 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-7 a^2 z^4+3 z^4+a^4 z^2+z^2-2 a^4+4 a^2-1}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 a^3 z^9+3 a z^9+7 a^4 z^8+13 a^2 z^8+6 z^8+8 a^5 z^7+4 a^3 z^7+4 z^7 a^{-1} +7 a^6 z^6-9 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -18 z^6+4 a^7 z^5-8 a^5 z^5-17 a^3 z^5-16 a z^5-11 z^5 a^{-1} +a^8 z^4-7 a^6 z^4+3 a^4 z^4+28 a^2 z^4-2 z^4 a^{-2} +15 z^4-3 a^7 z^3-a^5 z^3+9 a^3 z^3+13 a z^3+6 z^3 a^{-1} +a^6 z^2+a^4 z^2-3 a^2 z^2-3 z^2+2 a^5 z+2 a^3 z-2 a^4-4 a^2-1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a184,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 112"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-4 q^2+7 q-10+14 q^{-1} -14 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {K11a184,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(2, -2)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{220}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{68}{3}}

-128

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{736}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{160}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -48}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{256}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1760}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{544}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{14431}{15}}

-{\frac {604}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{21724}{45}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{545}{9}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1471}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or

j-2r=s-1

, where

s=

-2 is the signature of 10 112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

4

1

3

1

6

3

-3

-1

8

4

-3

7

0

-5

7

0

-7

4

7

3

-9

3

7

-4

-11

1

4

3

-13

3

-3

-15

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-4 q^9+2 q^8+14 q^7-23 q^6-8 q^5+54 q^4-41 q^3-47 q^2+106 q-36-103 q^{-1} +143 q^{-2} -12 q^{-3} -148 q^{-4} +148 q^{-5} +19 q^{-6} -158 q^{-7} +117 q^{-8} +38 q^{-9} -123 q^{-10} +66 q^{-11} +33 q^{-12} -65 q^{-13} +27 q^{-14} +14 q^{-15} -22 q^{-16} +9 q^{-17} +3 q^{-18} -4 q^{-19} + q^{-20} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-4 q^{20}+2 q^{19}+9 q^{18}-26 q^{16}-13 q^{15}+58 q^{14}+43 q^{13}-83 q^{12}-113 q^{11}+95 q^{10}+210 q^9-63 q^8-323 q^7-20 q^6+416 q^5+158 q^4-476 q^3-326 q^2+487 q+494-434 q^{-1} -666 q^{-2} +368 q^{-3} +785 q^{-4} -246 q^{-5} -906 q^{-6} +148 q^{-7} +956 q^{-8} -7 q^{-9} -1001 q^{-10} -100 q^{-11} +966 q^{-12} +231 q^{-13} -908 q^{-14} -310 q^{-15} +769 q^{-16} +375 q^{-17} -610 q^{-18} -380 q^{-19} +433 q^{-20} +332 q^{-21} -268 q^{-22} -259 q^{-23} +155 q^{-24} +163 q^{-25} -75 q^{-26} -94 q^{-27} +47 q^{-28} +37 q^{-29} -24 q^{-30} -19 q^{-31} +23 q^{-32} +3 q^{-33} -12 q^{-34} -2 q^{-35} +5 q^{-36} +3 q^{-37} -4 q^{-38} + q^{-39} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-4 q^{35}+2 q^{34}+9 q^{33}-5 q^{32}-3 q^{31}-32 q^{30}+11 q^{29}+70 q^{28}+16 q^{27}-6 q^{26}-192 q^{25}-83 q^{24}+210 q^{23}+235 q^{22}+230 q^{21}-476 q^{20}-576 q^{19}+21 q^{18}+537 q^{17}+1140 q^{16}-235 q^{15}-1220 q^{14}-1023 q^{13}+6 q^{12}+2270 q^{11}+1115 q^{10}-819 q^9-2284 q^8-1952 q^7+2231 q^6+2753 q^5+1209 q^4-2298 q^3-4366 q^2+491 q+3222+3856 q^{-1} -662 q^{-2} -5829 q^{-3} -1973 q^{-4} +2227 q^{-5} +5840 q^{-6} +1676 q^{-7} -6042 q^{-8} -4077 q^{-9} +585 q^{-10} +6878 q^{-11} +3792 q^{-12} -5519 q^{-13} -5562 q^{-14} -1089 q^{-15} +7191 q^{-16} +5500 q^{-17} -4440 q^{-18} -6449 q^{-19} -2802 q^{-20} +6586 q^{-21} +6689 q^{-22} -2562 q^{-23} -6251 q^{-24} -4374 q^{-25} +4635 q^{-26} +6680 q^{-27} -196 q^{-28} -4476 q^{-29} -4890 q^{-30} +1881 q^{-31} +4953 q^{-32} +1370 q^{-33} -1845 q^{-34} -3733 q^{-35} -122 q^{-36} +2403 q^{-37} +1347 q^{-38} -2 q^{-39} -1820 q^{-40} -576 q^{-41} +631 q^{-42} +533 q^{-43} +437 q^{-44} -530 q^{-45} -252 q^{-46} +51 q^{-47} +22 q^{-48} +230 q^{-49} -95 q^{-50} -17 q^{-51} -62 q^{-53} +59 q^{-54} -19 q^{-55} +17 q^{-56} +9 q^{-57} -23 q^{-58} +8 q^{-59} -6 q^{-60} +5 q^{-61} +3 q^{-62} -4 q^{-63} + q^{-64} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{55}-4 q^{54}+2 q^{53}+9 q^{52}-5 q^{51}-8 q^{50}-9 q^{49}-8 q^{48}+22 q^{47}+63 q^{46}+22 q^{45}-74 q^{44}-133 q^{43}-115 q^{42}+65 q^{41}+317 q^{40}+394 q^{39}+44 q^{38}-524 q^{37}-848 q^{36}-545 q^{35}+472 q^{34}+1534 q^{33}+1564 q^{32}+104 q^{31}-1937 q^{30}-2976 q^{29}-1751 q^{28}+1492 q^{27}+4402 q^{26}+4276 q^{25}+445 q^{24}-4706 q^{23}-7184 q^{22}-4190 q^{21}+3047 q^{20}+9219 q^{19}+9027 q^{18}+1320 q^{17}-8945 q^{16}-13720 q^{15}-8006 q^{14}+5306 q^{13}+16469 q^{12}+15821 q^{11}+1851 q^{10}-15820 q^9-22871 q^8-11687 q^7+11014 q^6+27476 q^5+22575 q^4-2495 q^3-28442 q^2-32575 q-8697+25475 q^{-1} +40565 q^{-2} +20767 q^{-3} -19455 q^{-4} -45362 q^{-5} -32362 q^{-6} +11130 q^{-7} +47609 q^{-8} +42408 q^{-9} -2411 q^{-10} -47264 q^{-11} -50446 q^{-12} -6411 q^{-13} +45857 q^{-14} +56709 q^{-15} +13975 q^{-16} -43492 q^{-17} -61496 q^{-18} -20995 q^{-19} +41233 q^{-20} +65425 q^{-21} +27032 q^{-22} -38415 q^{-23} -68688 q^{-24} -33287 q^{-25} +35262 q^{-26} +71176 q^{-27} +39436 q^{-28} -30398 q^{-29} -72336 q^{-30} -46097 q^{-31} +23873 q^{-32} +71274 q^{-33} +52009 q^{-34} -14921 q^{-35} -66993 q^{-36} -56686 q^{-37} +4579 q^{-38} +59125 q^{-39} +58319 q^{-40} +6144 q^{-41} -47851 q^{-42} -56204 q^{-43} -15438 q^{-44} +34582 q^{-45} +50059 q^{-46} +21632 q^{-47} -21187 q^{-48} -40637 q^{-49} -23983 q^{-50} +9608 q^{-51} +29845 q^{-52} +22500 q^{-53} -1315 q^{-54} -19347 q^{-55} -18472 q^{-56} -3367 q^{-57} +10885 q^{-58} +13263 q^{-59} +4988 q^{-60} -4934 q^{-61} -8507 q^{-62} -4597 q^{-63} +1606 q^{-64} +4668 q^{-65} +3401 q^{-66} +46 q^{-67} -2334 q^{-68} -2126 q^{-69} -429 q^{-70} +904 q^{-71} +1132 q^{-72} +495 q^{-73} -301 q^{-74} -557 q^{-75} -296 q^{-76} +49 q^{-77} +204 q^{-78} +169 q^{-79} +35 q^{-80} -72 q^{-81} -88 q^{-82} -15 q^{-83} +17 q^{-84} +12 q^{-85} +27 q^{-86} +3 q^{-87} -17 q^{-88} -3 q^{-89} +4 q^{-90} -6 q^{-91} +5 q^{-92} +3 q^{-93} -4 q^{-94} + q^{-95} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-4 q^{77}+2 q^{76}+9 q^{75}-5 q^{74}-8 q^{73}-14 q^{72}+15 q^{71}+3 q^{70}+15 q^{69}+68 q^{68}-26 q^{67}-86 q^{66}-153 q^{65}-21 q^{64}+56 q^{63}+210 q^{62}+491 q^{61}+190 q^{60}-250 q^{59}-903 q^{58}-836 q^{57}-620 q^{56}+367 q^{55}+2093 q^{54}+2348 q^{53}+1470 q^{52}-1324 q^{51}-3293 q^{50}-5067 q^{49}-3686 q^{48}+1697 q^{47}+6581 q^{46}+9489 q^{45}+6049 q^{44}-35 q^{43}-10547 q^{42}-16538 q^{41}-12353 q^{40}-932 q^{39}+14894 q^{38}+23370 q^{37}+23945 q^{36}+5342 q^{35}-18963 q^{34}-35541 q^{33}-35784 q^{32}-13872 q^{31}+17450 q^{30}+51592 q^{29}+54149 q^{28}+28028 q^{27}-18441 q^{26}-63016 q^{25}-78088 q^{24}-53589 q^{23}+17336 q^{22}+79351 q^{21}+108954 q^{20}+77740 q^{19}-4510 q^{18}-98846 q^{17}-152582 q^{16}-107807 q^{15}-3599 q^{14}+124933 q^{13}+192532 q^{12}+152823 q^{11}+7525 q^{10}-166458 q^9-241469 q^8-188271 q^7+217 q^6+206600 q^5+308987 q^4+214477 q^3-33219 q^2-265788 q-363557-220510 q^{-1} +75072 q^{-2} +353982 q^{-3} +406834 q^{-4} +188234 q^{-5} -153273 q^{-6} -431978 q^{-7} -423976 q^{-8} -133940 q^{-9} +275403 q^{-10} +501456 q^{-11} +391334 q^{-12} +24539 q^{-13} -392844 q^{-14} -541057 q^{-15} -323845 q^{-16} +142871 q^{-17} +505015 q^{-18} +519968 q^{-19} +183718 q^{-20} -310007 q^{-21} -582336 q^{-22} -450631 q^{-23} +25931 q^{-24} +473028 q^{-25} +587365 q^{-26} +292811 q^{-27} -238220 q^{-28} -595039 q^{-29} -531071 q^{-30} -57346 q^{-31} +445995 q^{-32} +634805 q^{-33} +374947 q^{-34} -181645 q^{-35} -607338 q^{-36} -604029 q^{-37} -139053 q^{-38} +414046 q^{-39} +680395 q^{-40} +469528 q^{-41} -97920 q^{-42} -595466 q^{-43} -678658 q^{-44} -259600 q^{-45} +324454 q^{-46} +686103 q^{-47} +575432 q^{-48} +52303 q^{-49} -497136 q^{-50} -701406 q^{-51} -403747 q^{-52} +142621 q^{-53} +581643 q^{-54} +621571 q^{-55} +235112 q^{-56} -287062 q^{-57} -596395 q^{-58} -481338 q^{-59} -75312 q^{-60} +355455 q^{-61} +529521 q^{-62} +342567 q^{-63} -44939 q^{-64} -369190 q^{-65} -415401 q^{-66} -208035 q^{-67} +108338 q^{-68} +322841 q^{-69} +304969 q^{-70} +102154 q^{-71} -135601 q^{-72} -246291 q^{-73} -198805 q^{-74} -36886 q^{-75} +122951 q^{-76} +177133 q^{-77} +115245 q^{-78} -4372 q^{-79} -91715 q^{-80} -112645 q^{-81} -61507 q^{-82} +17368 q^{-83} +65883 q^{-84} +64252 q^{-85} +24702 q^{-86} -15829 q^{-87} -40520 q^{-88} -34699 q^{-89} -8179 q^{-90} +14459 q^{-91} +22294 q^{-92} +14378 q^{-93} +2588 q^{-94} -9363 q^{-95} -12103 q^{-96} -5644 q^{-97} +1258 q^{-98} +5297 q^{-99} +4536 q^{-100} +2604 q^{-101} -1348 q^{-102} -3173 q^{-103} -1815 q^{-104} -316 q^{-105} +938 q^{-106} +949 q^{-107} +1020 q^{-108} -44 q^{-109} -699 q^{-110} -399 q^{-111} -182 q^{-112} +113 q^{-113} +98 q^{-114} +292 q^{-115} +43 q^{-116} -125 q^{-117} -49 q^{-118} -46 q^{-119} +12 q^{-120} -20 q^{-121} +58 q^{-122} +13 q^{-123} -23 q^{-124} +3 q^{-125} -7 q^{-126} +4 q^{-127} -6 q^{-128} +5 q^{-129} +3 q^{-130} -4 q^{-131} + q^{-132} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{105}-4 q^{104}+2 q^{103}+9 q^{102}-5 q^{101}-8 q^{100}-14 q^{99}+10 q^{98}+26 q^{97}-4 q^{96}+20 q^{95}+20 q^{94}-39 q^{93}-86 q^{92}-130 q^{91}-18 q^{90}+189 q^{89}+218 q^{88}+307 q^{87}+213 q^{86}-170 q^{85}-575 q^{84}-1114 q^{83}-937 q^{82}+19 q^{81}+1068 q^{80}+2323 q^{79}+2755 q^{78}+1680 q^{77}-476 q^{76}-4106 q^{75}-6650 q^{74}-6050 q^{73}-2726 q^{72}+4089 q^{71}+10866 q^{70}+14273 q^{69}+12544 q^{68}+2272 q^{67}-12339 q^{66}-24453 q^{65}-29328 q^{64}-19717 q^{63}+1745 q^{62}+27496 q^{61}+49490 q^{60}+51247 q^{59}+29082 q^{58}-11109 q^{57}-58147 q^{56}-86351 q^{55}-81674 q^{54}-40221 q^{53}+32573 q^{52}+103269 q^{51}+141541 q^{50}+125723 q^{49}+45576 q^{48}-66572 q^{47}-170667 q^{46}-221936 q^{45}-178291 q^{44}-51085 q^{43}+122230 q^{42}+276101 q^{41}+328024 q^{40}+248280 q^{39}+44859 q^{38}-219684 q^{37}-424559 q^{36}-479667 q^{35}-329249 q^{34}+1427 q^{33}+377537 q^{32}+650984 q^{31}+674467 q^{30}+385742 q^{29}-116839 q^{28}-651507 q^{27}-969702 q^{26}-876873 q^{25}-371829 q^{24}+388842 q^{23}+1079864 q^{22}+1348069 q^{21}+1026443 q^{20}+165462 q^{19}-892107 q^{18}-1646414 q^{17}-1716158 q^{16}-958013 q^{15}+358674 q^{14}+1638995 q^{13}+2276578 q^{12}+1862041 q^{11}+484452 q^{10}-1256298 q^9-2561391 q^8-2712791 q^7-1522784 q^6+513794 q^5+2479981 q^4+3353864 q^3+2601296 q^2+498263 q-2022845-3681823 q^{-1} -3565159 q^{-2} -1639523 q^{-3} +1255163 q^{-4} +3659966 q^{-5} +4297269 q^{-6} +2766712 q^{-7} -288310 q^{-8} -3324924 q^{-9} -4744622 q^{-10} -3758569 q^{-11} -745625 q^{-12} +2758223 q^{-13} +4907479 q^{-14} +4544110 q^{-15} +1737138 q^{-16} -2065024 q^{-17} -4840920 q^{-18} -5101061 q^{-19} -2600896 q^{-20} +1345575 q^{-21} +4615074 q^{-22} +5448137 q^{-23} +3301323 q^{-24} -677090 q^{-25} -4314371 q^{-26} -5632970 q^{-27} -3829595 q^{-28} +110075 q^{-29} +4003226 q^{-30} +5712096 q^{-31} +4213556 q^{-32} +338862 q^{-33} -3735338 q^{-34} -5745173 q^{-35} -4491455 q^{-36} -677796 q^{-37} +3532924 q^{-38} +5780653 q^{-39} +4718685 q^{-40} +940126 q^{-41} -3399370 q^{-42} -5851981 q^{-43} -4945050 q^{-44} -1177198 q^{-45} +3304370 q^{-46} +5969456 q^{-47} +5218300 q^{-48} +1449833 q^{-49} -3198548 q^{-50} -6111765 q^{-51} -5558109 q^{-52} -1818171 q^{-53} +3006066 q^{-54} +6225844 q^{-55} +5956977 q^{-56} +2321108 q^{-57} -2654449 q^{-58} -6228347 q^{-59} -6356639 q^{-60} -2958913 q^{-61} +2081549 q^{-62} +6023339 q^{-63} +6666957 q^{-64} +3679256 q^{-65} -1277096 q^{-66} -5532402 q^{-67} -6768298 q^{-68} -4374923 q^{-69} +287316 q^{-70} +4722091 q^{-71} +6558152 q^{-72} +4910362 q^{-73} +771904 q^{-74} -3634733 q^{-75} -5980942 q^{-76} -5151849 q^{-77} -1743975 q^{-78} +2384254 q^{-79} +5059882 q^{-80} +5018303 q^{-81} +2473724 q^{-82} -1138645 q^{-83} -3903117 q^{-84} -4509092 q^{-85} -2849932 q^{-86} +70836 q^{-87} +2672182 q^{-88} +3709833 q^{-89} +2847577 q^{-90} +692290 q^{-91} -1543588 q^{-92} -2768946 q^{-93} -2524928 q^{-94} -1100025 q^{-95} +650464 q^{-96} +1846579 q^{-97} +2004985 q^{-98} +1188097 q^{-99} -55431 q^{-100} -1072392 q^{-101} -1427253 q^{-102} -1050542 q^{-103} -255713 q^{-104} +512977 q^{-105} +906860 q^{-106} +801291 q^{-107} +350733 q^{-108} -169154 q^{-109} -508679 q^{-110} -538782 q^{-111} -318086 q^{-112} -1268 q^{-113} +247773 q^{-114} +321322 q^{-115} +232766 q^{-116} +60507 q^{-117} -99750 q^{-118} -171068 q^{-119} -147523 q^{-120} -62536 q^{-121} +30928 q^{-122} +81618 q^{-123} +81789 q^{-124} +44758 q^{-125} -4126 q^{-126} -35039 q^{-127} -41684 q^{-128} -26819 q^{-129} -2055 q^{-130} +14365 q^{-131} +19478 q^{-132} +13821 q^{-133} +2242 q^{-134} -5527 q^{-135} -8897 q^{-136} -6990 q^{-137} -1259 q^{-138} +2497 q^{-139} +4247 q^{-140} +3300 q^{-141} +370 q^{-142} -1068 q^{-143} -1866 q^{-144} -1680 q^{-145} -298 q^{-146} +445 q^{-147} +1047 q^{-148} +895 q^{-149} +11 q^{-150} -214 q^{-151} -358 q^{-152} -390 q^{-153} -94 q^{-154} + q^{-155} +207 q^{-156} +237 q^{-157} -17 q^{-158} -23 q^{-159} -45 q^{-160} -59 q^{-161} -9 q^{-162} -25 q^{-163} +26 q^{-164} +44 q^{-165} -13 q^{-166} -3 q^{-167} - q^{-168} -7 q^{-169} +4 q^{-170} -6 q^{-171} +5 q^{-172} +3 q^{-173} -4 q^{-174} + q^{-175} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

10_111

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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